statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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nontrivial.infsep_anti (hs : s.nontrivial) (hst : s ⊆ t) : t.infsep ≤ s.infsep | ennreal.to_real_mono hs.einfsep_ne_top (einfsep_anti hst) | lemma | set.nontrivial.infsep_anti | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"ennreal.to_real_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infsep_eq_infi [decidable s.nontrivial] :
s.infsep = if s.nontrivial then ⨅ d : s.off_diag, (uncurry dist) (d : α × α) else 0 | begin
split_ifs with hs,
{ have hb : bdd_below (uncurry dist '' s.off_diag),
{ refine ⟨0, λ d h, _⟩,
simp_rw [mem_image, prod.exists, uncurry_apply_pair] at h,
rcases h with ⟨_, _, _, rfl⟩,
exact dist_nonneg },
refine eq_of_forall_le_iff (λ _, _),
simp_rw [hs.le_infsep_iff, le_cinfi_se... | lemma | set.infsep_eq_infi | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"and_imp",
"bdd_below",
"dist_nonneg",
"eq_of_forall_le_iff",
"imp_forall_iff",
"le_cinfi_set_iff",
"uncurry_apply_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial.infsep_eq_infi (hs : s.nontrivial)
: s.infsep = ⨅ d : s.off_diag, (uncurry dist) (d : α × α) | by { classical, rw [infsep_eq_infi, if_pos hs] } | lemma | set.nontrivial.infsep_eq_infi | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infsep_of_fintype [decidable s.nontrivial] [decidable_eq α] [fintype s] :
s.infsep = if hs : s.nontrivial then s.off_diag.to_finset.inf' (by simpa) (uncurry dist) else 0 | begin
split_ifs with hs,
{ refine eq_of_forall_le_iff (λ _, _),
simp_rw [hs.le_infsep_iff, imp_forall_iff, finset.le_inf'_iff, mem_to_finset, mem_off_diag,
prod.forall, uncurry_apply_pair, and_imp] },
{ rw not_nontrivial_iff at hs, exact hs.infsep_zero }
end | lemma | set.infsep_of_fintype | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"and_imp",
"eq_of_forall_le_iff",
"finset.le_inf'_iff",
"fintype",
"imp_forall_iff",
"uncurry_apply_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial.infsep_of_fintype [decidable_eq α] [fintype s] (hs : s.nontrivial) :
s.infsep = s.off_diag.to_finset.inf' (by simpa) (uncurry dist) | by { classical, rw [infsep_of_fintype, dif_pos hs] } | lemma | set.nontrivial.infsep_of_fintype | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite.infsep [decidable s.nontrivial] (hsf : s.finite) :
s.infsep = if hs : s.nontrivial then hsf.off_diag.to_finset.inf' (by simpa) (uncurry dist)
else 0 | begin
split_ifs with hs,
{ refine eq_of_forall_le_iff (λ _, _),
simp_rw [hs.le_infsep_iff, imp_forall_iff, finset.le_inf'_iff, finite.mem_to_finset,
mem_off_diag, prod.forall, uncurry_apply_pair, and_imp] },
{ rw not_nontrivial_iff at hs, exact hs.infsep_zero }
end | lemma | set.finite.infsep | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"and_imp",
"eq_of_forall_le_iff",
"finset.le_inf'_iff",
"imp_forall_iff",
"uncurry_apply_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite.infsep_of_nontrivial (hsf : s.finite) (hs : s.nontrivial) :
s.infsep = hsf.off_diag.to_finset.inf' (by simpa) (uncurry dist) | by { classical, simp_rw [hsf.infsep, dif_pos hs] } | lemma | set.finite.infsep_of_nontrivial | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.finset.coe_infsep [decidable_eq α] (s : finset α) :
(s : set α).infsep = if hs : s.off_diag.nonempty then s.off_diag.inf' hs (uncurry dist)
else 0 | begin
have H : (s : set α).nontrivial ↔ s.off_diag.nonempty,
by rwa [← set.off_diag_nonempty, ← finset.coe_off_diag, finset.coe_nonempty],
split_ifs with hs,
{ simp_rw [(H.mpr hs).infsep_of_fintype, ← finset.coe_off_diag, finset.to_finset_coe] },
{ exact ((not_nontrivial_iff).mp (H.mp.mt hs)).infsep_zero }
en... | lemma | finset.coe_infsep | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"finset",
"finset.coe_nonempty",
"finset.coe_off_diag",
"finset.to_finset_coe",
"nontrivial",
"set.off_diag_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.finset.coe_infsep_of_off_diag_nonempty [decidable_eq α] {s : finset α}
(hs : s.off_diag.nonempty) : (s : set α).infsep = s.off_diag.inf' hs (uncurry dist) | by rw [finset.coe_infsep, dif_pos hs] | lemma | finset.coe_infsep_of_off_diag_nonempty | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"finset",
"finset.coe_infsep"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.finset.coe_infsep_of_off_diag_empty [decidable_eq α] {s : finset α}
(hs : s.off_diag = ∅) : (s : set α).infsep = 0 | by { rw ← finset.not_nonempty_iff_eq_empty at hs, rw [finset.coe_infsep, dif_neg hs] } | lemma | finset.coe_infsep_of_off_diag_empty | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"finset",
"finset.coe_infsep",
"finset.not_nonempty_iff_eq_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial.infsep_exists_of_finite [finite s] (hs : s.nontrivial) :
∃ (x y ∈ s) (hxy : x ≠ y), s.infsep = dist x y | begin
classical,
casesI nonempty_fintype s,
simp_rw hs.infsep_of_fintype,
rcases @finset.exists_mem_eq_inf' _ _ _ (s.off_diag.to_finset) (by simpa) (uncurry dist)
with ⟨_, hxy, hed⟩,
simp_rw mem_to_finset at hxy,
exact ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩
end | lemma | set.nontrivial.infsep_exists_of_finite | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"finite",
"finset.exists_mem_eq_inf'",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite.infsep_exists_of_nontrivial (hsf : s.finite) (hs : s.nontrivial) :
∃ (x y ∈ s) (hxy : x ≠ y), s.infsep = dist x y | by { letI := hsf.fintype, exact hs.infsep_exists_of_finite } | lemma | set.finite.infsep_exists_of_nontrivial | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infsep_zero_iff_subsingleton_of_finite [finite s] :
s.infsep = 0 ↔ s.subsingleton | begin
rw [infsep_zero, einfsep_eq_top_iff, or_iff_right_iff_imp],
exact λ H, (einfsep_pos_of_finite.ne' H).elim
end | lemma | set.infsep_zero_iff_subsingleton_of_finite | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"finite",
"or_iff_right_iff_imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infsep_pos_iff_nontrivial_of_finite [finite s] :
0 < s.infsep ↔ s.nontrivial | begin
rw [infsep_pos, einfsep_lt_top_iff, and_iff_right_iff_imp],
exact λ _, einfsep_pos_of_finite
end | lemma | set.infsep_pos_iff_nontrivial_of_finite | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"and_iff_right_iff_imp",
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite.infsep_zero_iff_subsingleton (hs : s.finite) :
s.infsep = 0 ↔ s.subsingleton | by { letI := hs.fintype, exact infsep_zero_iff_subsingleton_of_finite } | lemma | set.finite.infsep_zero_iff_subsingleton | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite.infsep_pos_iff_nontrivial (hs : s.finite) :
0 < s.infsep ↔ s.nontrivial | by { letI := hs.fintype, exact infsep_pos_iff_nontrivial_of_finite } | lemma | set.finite.infsep_pos_iff_nontrivial | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.finset.infsep_zero_iff_subsingleton (s : finset α) :
(s : set α).infsep = 0 ↔ (s : set α).subsingleton | infsep_zero_iff_subsingleton_of_finite | lemma | finset.infsep_zero_iff_subsingleton | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.finset.infsep_pos_iff_nontrivial (s : finset α) :
0 < (s : set α).infsep ↔ (s : set α).nontrivial | infsep_pos_iff_nontrivial_of_finite | lemma | finset.infsep_pos_iff_nontrivial | topology.metric_space | src/topology/metric_space/infsep.lean | [
"topology.metric_space.basic"
] | [
"finset",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_isometric_vadd [pseudo_emetric_space X] [has_vadd M X] : Prop | (isometry_vadd [] : ∀ c : M, isometry ((+ᵥ) c : X → X)) | class | has_isometric_vadd | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_vadd",
"isometry",
"pseudo_emetric_space"
] | An additive action is isometric if each map `x ↦ c +ᵥ x` is an isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_isometric_smul [pseudo_emetric_space X] [has_smul M X] : Prop | (isometry_smul [] : ∀ c : M, isometry ((•) c : X → X)) | class | has_isometric_smul | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_smul",
"isometry",
"pseudo_emetric_space"
] | A multiplicative action is isometric if each map `x ↦ c • x` is an isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_isometric_smul.to_has_continuous_const_smul [pseudo_emetric_space X] [has_smul M X]
[has_isometric_smul M X] : has_continuous_const_smul M X | ⟨λ c, (isometry_smul X c).continuous⟩ | instance | has_isometric_smul.to_has_continuous_const_smul | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_continuous_const_smul",
"has_isometric_smul",
"has_smul",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_isometric_smul.opposite_of_comm [pseudo_emetric_space X] [has_smul M X]
[has_smul Mᵐᵒᵖ X] [is_central_scalar M X] [has_isometric_smul M X] :
has_isometric_smul Mᵐᵒᵖ X | ⟨λ c x y, by simpa only [← op_smul_eq_smul] using (isometry_smul X c.unop x y)⟩ | instance | has_isometric_smul.opposite_of_comm | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"has_smul",
"is_central_scalar",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_smul_left [has_smul M X] [has_isometric_smul M X]
(c : M) (x y : X) :
edist (c • x) (c • y) = edist x y | isometry_smul X c x y | lemma | edist_smul_left | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_smul [has_smul M X] [has_isometric_smul M X] (c : M) (s : set X) :
emetric.diam (c • s) = emetric.diam s | (isometry_smul _ _).ediam_image s | lemma | ediam_smul | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"emetric.diam",
"has_isometric_smul",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry_mul_left [has_mul M] [pseudo_emetric_space M]
[has_isometric_smul M M] (a : M) : isometry ((*) a) | isometry_smul M a | lemma | isometry_mul_left | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"isometry",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_mul_left [has_mul M] [pseudo_emetric_space M]
[has_isometric_smul M M] (a b c : M) : edist (a * b) (a * c) = edist b c | isometry_mul_left a b c | lemma | edist_mul_left | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"isometry_mul_left",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry_mul_right [has_mul M] [pseudo_emetric_space M]
[has_isometric_smul Mᵐᵒᵖ M] (a : M) : isometry (λ x, x * a) | isometry_smul M (mul_opposite.op a) | lemma | isometry_mul_right | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"isometry",
"mul_opposite.op",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_mul_right [has_mul M] [pseudo_emetric_space M]
[has_isometric_smul Mᵐᵒᵖ M] (a b c : M) : edist (a * c) (b * c) = edist a b | isometry_mul_right c a b | lemma | edist_mul_right | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"isometry_mul_right",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_div_right [div_inv_monoid M] [pseudo_emetric_space M]
[has_isometric_smul Mᵐᵒᵖ M] (a b c : M) : edist (a / c) (b / c) = edist a b | by simp only [div_eq_mul_inv, edist_mul_right] | lemma | edist_div_right | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"div_eq_mul_inv",
"div_inv_monoid",
"edist_mul_right",
"has_isometric_smul",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_inv_inv [pseudo_emetric_space G] [has_isometric_smul G G]
[has_isometric_smul Gᵐᵒᵖ G] (a b : G) : edist a⁻¹ b⁻¹ = edist a b | by rw [← edist_mul_left a, ← edist_mul_right _ _ b, mul_right_inv, one_mul,
inv_mul_cancel_right, edist_comm] | lemma | edist_inv_inv | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"edist_mul_left",
"edist_mul_right",
"has_isometric_smul",
"inv_mul_cancel_right",
"mul_right_inv",
"one_mul",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry_inv [pseudo_emetric_space G] [has_isometric_smul G G]
[has_isometric_smul Gᵐᵒᵖ G] : isometry (has_inv.inv : G → G) | edist_inv_inv | lemma | isometry_inv | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"edist_inv_inv",
"has_isometric_smul",
"isometry",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_inv [pseudo_emetric_space G] [has_isometric_smul G G]
[has_isometric_smul Gᵐᵒᵖ G] (x y : G) : edist x⁻¹ y = edist x y⁻¹ | by rw [← edist_inv_inv, inv_inv] | lemma | edist_inv | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"edist_inv_inv",
"has_isometric_smul",
"inv_inv",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_div_left [pseudo_emetric_space G] [has_isometric_smul G G]
[has_isometric_smul Gᵐᵒᵖ G] (a b c : G) : edist (a / b) (a / c) = edist b c | by rw [div_eq_mul_inv, div_eq_mul_inv, edist_mul_left, edist_inv_inv] | lemma | edist_div_left | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"div_eq_mul_inv",
"edist_inv_inv",
"edist_mul_left",
"has_isometric_smul",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_smul (c : G) : X ≃ᵢ X | { to_equiv := mul_action.to_perm c,
isometry_to_fun := isometry_smul X c } | def | isometry_equiv.const_smul | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"mul_action.to_perm"
] | If a group `G` acts on `X` by isometries, then `isometry_equiv.const_smul` is the isometry of
`X` given by multiplication of a constant element of the group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_smul_symm (c : G) : (const_smul c : X ≃ᵢ X).symm = const_smul c⁻¹ | ext $ λ _, rfl | lemma | isometry_equiv.const_smul_symm | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left [has_isometric_smul G G] (c : G) : G ≃ᵢ G | { to_equiv := equiv.mul_left c,
isometry_to_fun := edist_mul_left c } | def | isometry_equiv.mul_left | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"edist_mul_left",
"equiv.mul_left",
"has_isometric_smul"
] | Multiplication `y ↦ x * y` as an `isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_left_symm [has_isometric_smul G G] (x : G) :
(mul_left x).symm = isometry_equiv.mul_left x⁻¹ | const_smul_symm x | lemma | isometry_equiv.mul_left_symm | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"isometry_equiv.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right [has_isometric_smul Gᵐᵒᵖ G] (c : G) : G ≃ᵢ G | { to_equiv := equiv.mul_right c,
isometry_to_fun := λ a b, edist_mul_right a b c } | def | isometry_equiv.mul_right | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"edist_mul_right",
"equiv.mul_right",
"has_isometric_smul"
] | Multiplication `y ↦ y * x` as an `isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_right_symm [has_isometric_smul Gᵐᵒᵖ G] (x : G) :
(mul_right x).symm = mul_right x⁻¹ | ext $ λ y, rfl | lemma | isometry_equiv.mul_right_symm | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_right [has_isometric_smul Gᵐᵒᵖ G] (c : G) : G ≃ᵢ G | { to_equiv := equiv.div_right c,
isometry_to_fun := λ a b, edist_div_right a b c } | def | isometry_equiv.div_right | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"edist_div_right",
"equiv.div_right",
"has_isometric_smul"
] | Division `y ↦ y / x` as an `isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_right_symm [has_isometric_smul Gᵐᵒᵖ G] (c : G) :
(div_right c).symm = mul_right c | ext $ λ y, rfl | lemma | isometry_equiv.div_right_symm | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_left (c : G) : G ≃ᵢ G | { to_equiv := equiv.div_left c,
isometry_to_fun := edist_div_left c } | def | isometry_equiv.div_left | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"edist_div_left",
"equiv.div_left"
] | Division `y ↦ x / y` as an `isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv : G ≃ᵢ G | { to_equiv := equiv.inv G,
isometry_to_fun := edist_inv_inv } | def | isometry_equiv.inv | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"edist_inv_inv",
"equiv.inv"
] | Inversion `x ↦ x⁻¹` as an `isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_symm : (inv G).symm = inv G | rfl | lemma | isometry_equiv.inv_symm | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_ball (c : G) (x : X) (r : ℝ≥0∞) :
c • ball x r = ball (c • x) r | (isometry_equiv.const_smul c).image_emetric_ball _ _ | lemma | emetric.smul_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"isometry_equiv.const_smul",
"smul_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_smul_ball (c : G) (x : X) (r : ℝ≥0∞) :
((•) c) ⁻¹' ball x r = ball (c⁻¹ • x) r | by rw [preimage_smul, smul_ball] | lemma | emetric.preimage_smul_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"smul_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_closed_ball (c : G) (x : X) (r : ℝ≥0∞) :
c • closed_ball x r = closed_ball (c • x) r | (isometry_equiv.const_smul c).image_emetric_closed_ball _ _ | lemma | emetric.smul_closed_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"isometry_equiv.const_smul",
"smul_closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_smul_closed_ball (c : G) (x : X) (r : ℝ≥0∞) :
((•) c) ⁻¹' closed_ball x r = closed_ball (c⁻¹ • x) r | by rw [preimage_smul, smul_closed_ball] | lemma | emetric.preimage_smul_closed_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"smul_closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_mul_left_ball [has_isometric_smul G G] (a b : G) (r : ℝ≥0∞) :
((*) a) ⁻¹' ball b r = ball (a⁻¹ * b) r | preimage_smul_ball a b r | lemma | emetric.preimage_mul_left_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_mul_right_ball [has_isometric_smul Gᵐᵒᵖ G] (a b : G) (r : ℝ≥0∞) :
(λ x, x * a) ⁻¹' ball b r = ball (b / a) r | by { rw div_eq_mul_inv, exact preimage_smul_ball (mul_opposite.op a) b r } | lemma | emetric.preimage_mul_right_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"div_eq_mul_inv",
"has_isometric_smul",
"mul_opposite.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_mul_left_closed_ball [has_isometric_smul G G] (a b : G) (r : ℝ≥0∞) :
((*) a) ⁻¹' closed_ball b r = closed_ball (a⁻¹ * b) r | preimage_smul_closed_ball a b r | lemma | emetric.preimage_mul_left_closed_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_mul_right_closed_ball [has_isometric_smul Gᵐᵒᵖ G] (a b : G) (r : ℝ≥0∞) :
(λ x, x * a) ⁻¹' closed_ball b r = closed_ball (b / a) r | by { rw div_eq_mul_inv, exact preimage_smul_closed_ball (mul_opposite.op a) b r } | lemma | emetric.preimage_mul_right_closed_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"div_eq_mul_inv",
"has_isometric_smul",
"mul_opposite.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_smul [pseudo_metric_space X] [has_smul M X] [has_isometric_smul M X]
(c : M) (x y : X) : dist (c • x) (c • y) = dist x y | (isometry_smul X c).dist_eq x y | lemma | dist_smul | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"has_smul",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_smul [pseudo_metric_space X] [has_smul M X] [has_isometric_smul M X]
(c : M) (x y : X) : nndist (c • x) (c • y) = nndist x y | (isometry_smul X c).nndist_eq x y | lemma | nndist_smul | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"has_smul",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_smul [pseudo_metric_space X] [has_smul M X] [has_isometric_smul M X]
(c : M) (s : set X) : metric.diam (c • s) = metric.diam s | (isometry_smul _ _).diam_image s | lemma | diam_smul | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"has_smul",
"metric.diam",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_mul_left [pseudo_metric_space M] [has_mul M] [has_isometric_smul M M]
(a b c : M) : dist (a * b) (a * c) = dist b c | dist_smul a b c | lemma | dist_mul_left | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"dist_smul",
"has_isometric_smul",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_mul_left [pseudo_metric_space M] [has_mul M] [has_isometric_smul M M]
(a b c : M) : nndist (a * b) (a * c) = nndist b c | nndist_smul a b c | lemma | nndist_mul_left | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"nndist_smul",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_mul_right [has_mul M] [pseudo_metric_space M]
[has_isometric_smul Mᵐᵒᵖ M] (a b c : M) : dist (a * c) (b * c) = dist a b | dist_smul (mul_opposite.op c) a b | lemma | dist_mul_right | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"dist_smul",
"has_isometric_smul",
"mul_opposite.op",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_mul_right [pseudo_metric_space M] [has_mul M] [has_isometric_smul Mᵐᵒᵖ M]
(a b c : M) : nndist (a * c) (b * c) = nndist a b | nndist_smul (mul_opposite.op c) a b | lemma | nndist_mul_right | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"mul_opposite.op",
"nndist_smul",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_div_right [div_inv_monoid M] [pseudo_metric_space M]
[has_isometric_smul Mᵐᵒᵖ M] (a b c : M) : dist (a / c) (b / c) = dist a b | by simp only [div_eq_mul_inv, dist_mul_right] | lemma | dist_div_right | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"dist_mul_right",
"div_eq_mul_inv",
"div_inv_monoid",
"has_isometric_smul",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_div_right [div_inv_monoid M] [pseudo_metric_space M]
[has_isometric_smul Mᵐᵒᵖ M] (a b c : M) : nndist (a / c) (b / c) = nndist a b | by simp only [div_eq_mul_inv, nndist_mul_right] | lemma | nndist_div_right | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"div_eq_mul_inv",
"div_inv_monoid",
"has_isometric_smul",
"nndist_mul_right",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_inv_inv [group G] [pseudo_metric_space G] [has_isometric_smul G G]
[has_isometric_smul Gᵐᵒᵖ G] (a b : G) : dist a⁻¹ b⁻¹ = dist a b | (isometry_equiv.inv G).dist_eq a b | lemma | dist_inv_inv | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"group",
"has_isometric_smul",
"isometry_equiv.inv",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_inv_inv [group G] [pseudo_metric_space G] [has_isometric_smul G G]
[has_isometric_smul Gᵐᵒᵖ G] (a b : G) : nndist a⁻¹ b⁻¹ = nndist a b | (isometry_equiv.inv G).nndist_eq a b | lemma | nndist_inv_inv | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"group",
"has_isometric_smul",
"isometry_equiv.inv",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_div_left [group G] [pseudo_metric_space G] [has_isometric_smul G G]
[has_isometric_smul Gᵐᵒᵖ G] (a b c : G) : dist (a / b) (a / c) = dist b c | by simp [div_eq_mul_inv] | lemma | dist_div_left | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"div_eq_mul_inv",
"group",
"has_isometric_smul",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_div_left [group G] [pseudo_metric_space G] [has_isometric_smul G G]
[has_isometric_smul Gᵐᵒᵖ G] (a b c : G) : nndist (a / b) (a / c) = nndist b c | by simp [div_eq_mul_inv] | lemma | nndist_div_left | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"div_eq_mul_inv",
"group",
"has_isometric_smul",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_ball (c : G) (x : X) (r : ℝ) :
c • ball x r = ball (c • x) r | (isometry_equiv.const_smul c).image_ball _ _ | lemma | metric.smul_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"isometry_equiv.const_smul",
"smul_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_smul_ball (c : G) (x : X) (r : ℝ) :
((•) c) ⁻¹' ball x r = ball (c⁻¹ • x) r | by rw [preimage_smul, smul_ball] | lemma | metric.preimage_smul_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"smul_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_closed_ball (c : G) (x : X) (r : ℝ) :
c • closed_ball x r = closed_ball (c • x) r | (isometry_equiv.const_smul c).image_closed_ball _ _ | lemma | metric.smul_closed_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"isometry_equiv.const_smul",
"smul_closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_smul_closed_ball (c : G) (x : X) (r : ℝ) :
((•) c) ⁻¹' closed_ball x r = closed_ball (c⁻¹ • x) r | by rw [preimage_smul, smul_closed_ball] | lemma | metric.preimage_smul_closed_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"smul_closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_sphere (c : G) (x : X) (r : ℝ) :
c • sphere x r = sphere (c • x) r | (isometry_equiv.const_smul c).image_sphere _ _ | lemma | metric.smul_sphere | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"isometry_equiv.const_smul",
"smul_sphere"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_smul_sphere (c : G) (x : X) (r : ℝ) :
((•) c) ⁻¹' sphere x r = sphere (c⁻¹ • x) r | by rw [preimage_smul, smul_sphere] | lemma | metric.preimage_smul_sphere | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"smul_sphere"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_mul_left_ball [has_isometric_smul G G] (a b : G) (r : ℝ) :
((*) a) ⁻¹' ball b r = ball (a⁻¹ * b) r | preimage_smul_ball a b r | lemma | metric.preimage_mul_left_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_mul_right_ball [has_isometric_smul Gᵐᵒᵖ G] (a b : G) (r : ℝ) :
(λ x, x * a) ⁻¹' ball b r = ball (b / a) r | by { rw div_eq_mul_inv, exact preimage_smul_ball (mul_opposite.op a) b r } | lemma | metric.preimage_mul_right_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"div_eq_mul_inv",
"has_isometric_smul",
"mul_opposite.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_mul_left_closed_ball [has_isometric_smul G G] (a b : G) (r : ℝ) :
((*) a) ⁻¹' closed_ball b r = closed_ball (a⁻¹ * b) r | preimage_smul_closed_ball a b r | lemma | metric.preimage_mul_left_closed_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_mul_right_closed_ball [has_isometric_smul Gᵐᵒᵖ G] (a b : G) (r : ℝ) :
(λ x, x * a) ⁻¹' closed_ball b r = closed_ball (b / a) r | by { rw div_eq_mul_inv, exact preimage_smul_closed_ball (mul_opposite.op a) b r } | lemma | metric.preimage_mul_right_closed_ball | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"div_eq_mul_inv",
"has_isometric_smul",
"mul_opposite.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.has_isometric_smul' {N}
[has_mul M] [pseudo_emetric_space M] [has_isometric_smul M M]
[has_mul N] [pseudo_emetric_space N] [has_isometric_smul N N] :
has_isometric_smul (M × N) (M × N) | ⟨λ c, (isometry_smul M c.1).prod_map (isometry_smul N c.2)⟩ | instance | prod.has_isometric_smul' | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"prod_map",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.has_isometric_smul'' {N}
[has_mul M] [pseudo_emetric_space M] [has_isometric_smul Mᵐᵒᵖ M]
[has_mul N] [pseudo_emetric_space N] [has_isometric_smul Nᵐᵒᵖ N] :
has_isometric_smul (M × N)ᵐᵒᵖ (M × N) | ⟨λ c, (isometry_mul_right c.unop.1).prod_map (isometry_mul_right c.unop.2)⟩ | instance | prod.has_isometric_smul'' | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"isometry_mul_right",
"prod_map",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.has_isometric_smul [monoid M] : has_isometric_smul Mˣ X | ⟨λ c, by convert isometry_smul X (c : M)⟩ | instance | units.has_isometric_smul | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift.has_isometric_smul : has_isometric_smul (ulift M) X | ⟨λ c, by simpa only using isometry_smul X c.down⟩ | instance | ulift.has_isometric_smul | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift.has_isometric_smul' : has_isometric_smul M (ulift X) | ⟨λ c x y, by simpa only using edist_smul_left c x.1 y.1⟩ | instance | ulift.has_isometric_smul' | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"edist_smul_left",
"has_isometric_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.has_isometric_smul' {ι} {M X : ι → Type*} [fintype ι]
[Π i, has_smul (M i) (X i)] [Π i, pseudo_emetric_space (X i)]
[∀ i, has_isometric_smul (M i) (X i)] :
has_isometric_smul (Π i, M i) (Π i, X i) | ⟨λ c, isometry_dcomp (λ i, (•) (c i)) (λ i, isometry_smul _ _)⟩ | instance | pi.has_isometric_smul' | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"fintype",
"has_isometric_smul",
"has_smul",
"isometry_dcomp",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.has_isometric_smul'' {ι} {M : ι → Type*} [fintype ι]
[Π i, has_mul (M i)] [Π i, pseudo_emetric_space (M i)] [∀ i, has_isometric_smul (M i)ᵐᵒᵖ (M i)] :
has_isometric_smul (Π i, M i)ᵐᵒᵖ (Π i, M i) | ⟨λ c, isometry_dcomp (λ i (x : M i), x * c.unop i) $ λ i, isometry_mul_right _⟩ | instance | pi.has_isometric_smul'' | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"fintype",
"has_isometric_smul",
"isometry_dcomp",
"isometry_mul_right",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
additive.has_isometric_vadd : has_isometric_vadd (additive M) X | ⟨λ c, isometry_smul X c.to_mul⟩ | instance | additive.has_isometric_vadd | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"additive",
"has_isometric_vadd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
additive.has_isometric_vadd' [has_mul M] [pseudo_emetric_space M]
[has_isometric_smul M M] : has_isometric_vadd (additive M) (additive M) | ⟨λ c x y, edist_smul_left c.to_mul x.to_mul y.to_mul⟩ | instance | additive.has_isometric_vadd' | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"additive",
"edist_smul_left",
"has_isometric_smul",
"has_isometric_vadd",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
additive.has_isometric_vadd'' [has_mul M] [pseudo_emetric_space M]
[has_isometric_smul Mᵐᵒᵖ M] : has_isometric_vadd (additive M)ᵃᵒᵖ (additive M) | ⟨λ c x y, edist_smul_left (mul_opposite.op c.unop.to_mul) x.to_mul y.to_mul⟩ | instance | additive.has_isometric_vadd'' | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"additive",
"edist_smul_left",
"has_isometric_smul",
"has_isometric_vadd",
"mul_opposite.op",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicative.has_isometric_smul {M X} [has_vadd M X] [pseudo_emetric_space X]
[has_isometric_vadd M X]: has_isometric_smul (multiplicative M) X | ⟨λ c, isometry_vadd X c.to_add⟩ | instance | multiplicative.has_isometric_smul | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"has_isometric_vadd",
"has_vadd",
"multiplicative",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicative.has_isometric_smul' [has_add M] [pseudo_emetric_space M]
[has_isometric_vadd M M] : has_isometric_smul (multiplicative M) (multiplicative M) | ⟨λ c x y, edist_vadd_left c.to_add x.to_add y.to_add⟩ | instance | multiplicative.has_isometric_smul' | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"has_isometric_vadd",
"multiplicative",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicative.has_isometric_vadd'' [has_add M] [pseudo_emetric_space M]
[has_isometric_vadd Mᵃᵒᵖ M] :
has_isometric_smul (multiplicative M)ᵐᵒᵖ (multiplicative M) | ⟨λ c x y, edist_vadd_left (add_opposite.op c.unop.to_add) x.to_add y.to_add⟩ | instance | multiplicative.has_isometric_vadd'' | topology.metric_space | src/topology/metric_space/isometric_smul.lean | [
"topology.metric_space.isometry"
] | [
"has_isometric_smul",
"has_isometric_vadd",
"multiplicative",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry [pseudo_emetric_space α] [pseudo_emetric_space β] (f : α → β) : Prop | ∀x1 x2 : α, edist (f x1) (f x2) = edist x1 x2 | def | isometry | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"pseudo_emetric_space"
] | An isometry (also known as isometric embedding) is a map preserving the edistance
between pseudoemetric spaces, or equivalently the distance between pseudometric space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isometry_iff_nndist_eq [pseudo_metric_space α] [pseudo_metric_space β] {f : α → β} :
isometry f ↔ (∀x y, nndist (f x) (f y) = nndist x y) | by simp only [isometry, edist_nndist, ennreal.coe_eq_coe] | lemma | isometry_iff_nndist_eq | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"edist_nndist",
"ennreal.coe_eq_coe",
"isometry",
"pseudo_metric_space"
] | On pseudometric spaces, a map is an isometry if and only if it preserves nonnegative
distances. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isometry_iff_dist_eq [pseudo_metric_space α] [pseudo_metric_space β] {f : α → β} :
isometry f ↔ (∀x y, dist (f x) (f y) = dist x y) | by simp only [isometry_iff_nndist_eq, ← coe_nndist, nnreal.coe_eq] | lemma | isometry_iff_dist_eq | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"coe_nndist",
"isometry",
"isometry_iff_nndist_eq",
"nnreal.coe_eq",
"pseudo_metric_space"
] | On pseudometric spaces, a map is an isometry if and only if it preserves distances. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_eq (hf : isometry f) (x y : α) : edist (f x) (f y) = edist x y | hf x y | theorem | isometry.edist_eq | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry"
] | An isometry preserves edistances. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz (h : isometry f) : lipschitz_with 1 f | lipschitz_with.of_edist_le $ λ x y, (h x y).le | lemma | isometry.lipschitz | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry",
"lipschitz_with",
"lipschitz_with.of_edist_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz (h : isometry f) : antilipschitz_with 1 f | λ x y, by simp only [h x y, ennreal.coe_one, one_mul, le_refl] | lemma | isometry.antilipschitz | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"antilipschitz_with",
"ennreal.coe_one",
"isometry",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.isometry_subsingleton [subsingleton α] : isometry f | λx y, by rw subsingleton.elim x y; simp | theorem | isometry_subsingleton | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry"
] | Any map on a subsingleton is an isometry | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.isometry_id : isometry (id : α → α) | λ x y, rfl | lemma | isometry_id | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry"
] | The identity is an isometry | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_map {δ} [pseudo_emetric_space δ] {f : α → β} {g : γ → δ} (hf : isometry f)
(hg : isometry g) : isometry (prod.map f g) | λ x y, by simp only [prod.edist_eq, hf.edist_eq, hg.edist_eq, prod_map] | lemma | isometry.prod_map | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry",
"prod.edist_eq",
"prod_map",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.isometry_dcomp {ι} [fintype ι] {α β : ι → Type*} [Π i, pseudo_emetric_space (α i)]
[Π i, pseudo_emetric_space (β i)] (f : Π i, α i → β i) (hf : ∀ i, isometry (f i)) :
isometry (dcomp f) | λ x y, by simp only [edist_pi_def, (hf _).edist_eq] | lemma | isometry_dcomp | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"edist_pi_def",
"fintype",
"isometry",
"pseudo_emetric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {g : β → γ} {f : α → β} (hg : isometry g) (hf : isometry f) : isometry (g ∘ f) | λ x y, (hg _ _).trans (hf _ _) | theorem | isometry.comp | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry"
] | The composition of isometries is an isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous (hf : isometry f) : uniform_continuous f | hf.lipschitz.uniform_continuous | theorem | isometry.uniform_continuous | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry",
"uniform_continuous"
] | An isometry from a metric space is a uniform continuous map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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